DYNAMICS FORCED BY HOMOCLINIC ORBITS

VALENT´IN MENDOZA

Abstract. The complexity of a dynamical system exhibiting a homoclinic orbit is given by the orbits that it forces. In this work we present a method, based in pruning theory, to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the 2-disk. Due to Cantwell and Conlon, this set is uniquely determined in the isotopy class of the orbit, up a topological conjugacy, so it contains the dynamics forced by the homoclinic orbit. Moreover we apply the method for finding the orbits forced by certain infinite families of homoclinic horseshoe orbits and propose its generalization to an arbitrary Smale map.

1. Introduction

Since the Poincar´e’sdiscovery of homoclinic orbits, it is known that dynamical systems with one of these orbits have a very complex behaviour. Such a feature was explained by Smale in terms of his celebrated horseshoe map [40]; more precisely, if a surface diffeomorphism f has a homoclinic point then, there exists an invariant set Λ where a power of f is conjugated to the shift σ defined on the

compact space Σ2 = {0, 1}Z of symbol sequences. To understand how complex is a diffeomorphism having a periodic or homoclinic orbit, we need the notion of forcing. First we will define it for periodic orbits.

1.1. Braid types of periodic orbits. Let P and Q be two periodic orbits of homeomorphisms f and g of the closed disk D2, respectively. We say that (P, f) and (Q, g) are equivalent if there is an orientation-preserving homeomorphism h : D2 → D2 with h(P ) = Q such that f is isotopic to h−1 ◦ g ◦ h relative to P . The equivalence class containing (P, f) is called the braid type bt(P, f). When the homeomorphism f is fixed, it will be written bt(P ) instead of bt(P, f). Now we can define the forcing relation between two braid types β and γ. We say that β forces γ, 2 denoted by β >2 γ, if every homeomorphism of D which has an orbit with braid type β, exhibits also an orbit with braid type γ. So we say that P forces Q, denoted by P >2 Q, if bt(P ) >2 bt(Q). In [6] Boyland proved that >2 is a partial order. If E(P ) is the set of periodic points whose orbits are forced by P , one can ensure that the dynamics of a diffeomorphism f containing an orbit with the braid arXiv:1412.2737v5 [math.DS] 23 Apr 2019 type of P is at least as complicated as f is restricted to E(P ). To find E(P ) it is necessary to use the Nielsen-Thurston theory of classification of surface homeomorphisms up to isotopies. In fact, Asimov and Franks [3] and Hall [25] showed that if f is pseudo-Anosov on D2 \ P then E(P ) is formed by the periodic points of its canonical representative φ. If P is of reducible type then the Nielsen-Thurston representative φ does not have the minimal number of periodic orbits, but a refinement of φ, due to Boyland [6,7], is used to get a condensed map which satisfies that property. An interesting case is when P is a periodic orbit of a Smale map f on D2, that is, f is an Axiom A map with a strong transversality between the invariant manifolds of its non-wandering points. In this

Date: October 30, 2015. 2010 Mathematics Subject Classification. Primary 37E30, 37E15, 37C29. Secondary 37B10, 37D20. Key words and phrases. Homoclinic orbits, braid types, dynamical core, Smale horseshoe, pruning theory. 1 2 VALENT´IN MENDOZA context the notion of bigon has became a tool for finding the Nielsen-Thurston representative rel to P : a bigon is a simply connected open region disjoint from P which is bounded by a stable segment and an unstable segment. In fact, Bonatti and Jeandenans [5] have proved that if there are no wandering bigons rel to a periodic orbit P of a transitive basic piece K, there exists a pseudo-Anosov map φ, with possibly 1-pronged singularities, and a semiconjugacy π between f and φ which is injective on the set of periodic orbits of K except on the boundary ones. If we do not admit non-wandering bigons too, the Bonatti-Jeandenans map φ is actually pseudo-Anosov and then the non-boundary periodic orbits of K are all forced by P . Thus the non-existence of bigons implies that the periodic orbits of K are forced by P . A similar result was obtained by Lewowicz and Ures in [33]: if f is a Smale map without bigons relative to P , which is called exteriorly situated, then its basic set is contained in the persistent set given by Handel [27] relative to P . More precisely it follows from Grines [23, 22] that if the non-wandering set of a Smale map contains an exteriorly situated basic set that does not contain special pairs of boundary periodic point (definition6) then there exists a semiconjugacy between f and a hyperbolic homeomorphism f0 which can be considered the canonical hyperbolic representative of the homotopy class of f. See [2, 24]. Since there exists a decomposition in basic pieces for Smale maps, the same analysis can even be applied if f has several transitive pieces and does not have bigons rel to P , considering the return maps to each basic piece. It will be the case, for example, if P is a renormalization of two or more orbits of pseudo-Anosov type [35].

1.2. Forcing on homoclinic orbits. Let us now suppose that P is a homoclinic orbit to a fixed point of a Smale diffeomorphism. In that case there are substantial differences but also useful similarities with the periodic case. First in [34] Los has proved that the forcing relation on periodic orbits can be extended to homoclinic or heteroclinic orbits in a suitable topology. There are several methods for finding the dynamics forced by a homoclinic orbit of a diffeomorphism f. For example, in [11, 12], Collins has constructed surface hyperbolic diffeomorphisms associated to a homoclinic orbit which can be used for approximating the entropy of that orbit as close as we want. His method uses the homoclinic tangle associated to the orbit. In [41, 43], Yamaguchi and Tanikawa have studied the forcing relation of reversible homoclinic horseshoe orbits appearing in area-preserving H´enonmaps. In other direction Boyland and Hall have given conditions for which a periodic orbit is isotopy stable relative to a compact set [8], and their result can be used for studying homoclinic orbits. 2 Since f restricted to MP := Int(D ) \ P is isotopic to an end periodic homeomorphism (lemma8), it is more convenient to use the Handel-Miller theory of classification of end periodic automorphisms on surfaces [29] as is presented in [20]. In this case Cantwell and Conlon [10] have proved that there exist a Handel-Miller map h and a set CP , called dynamical core, which is the intersection of the pair of totally disconnected and transverse geodesic laminations, such that h: C → C is uniquely determined by the isotopy class of f on MP . Thus one can say as definition that CP is the set of orbits forced by

P or that the dynamics of h on CP is forced by P : we say that an (finite or infinite) h-orbit Q is forced by P , denoted by P >2 Q, if Q ⊂ CP . In the general case, if an end periodic map f is irreducible [20], the union of the laminations fills M and f is isotopic to a pseudo-Anosov-like representative of the dynamics rel to P which preserves a pair of geodesic laminations. In our case, as in the work of Grines [23] for surfaces of finite topology, one has the following result (theorem 14). DYNAMICS FORCED BY HOMOCLINIC ORBITS 3

Theorem 1. Let f be a Smale map with a exteriorly situated basic set K without special pairs rel to an homoclinic orbit P . Then CP = K up a topological finite-to-one semi-conjugacy ι : CP → K which is injective on the set of non-boundary periodic points.

Therefore, in the hypothesis of theorem1, K contains the orbits forced by P .

1.3. Results of the paper. This is the context where our work is inserted. An important element of this paper is the pruning theory introduced by de Carvalho in [14] which is a technique for eliminating dynamics of a surface homeomorphism. So in section2 we state the differentiable pruning theorem (theorem 18), due to de Carvalho and the author, that can be used to eliminate bigons of a Smale map and hence for finding the dynamical core associate to a homoclinic orbit. This differentiable version of pruning have been implemented for uncrossing invariant manifolds of a Smale diffeomorphism in a region called a pruning domain. This point of view was inspired by the work of P. Cvitanovi´c[13] where a generic one-folding map is interpreted as a partial horseshoe, that is, a map whose dynamics forbids or prunes certain horseshoe orbits. As an application we will study, in section3, the forcing relation of homoclinic orbits coming from the standard Smale horseshoe F stated in [40]. In particular, it will be dealt homoclinic orbits to the ∞ w ∞ 1 1 ∞ fixed point 0 , that is, orbits P0 whose code in Σ2 is 010w0 · 10 where w is a finite word called w decoration. In [16] de Carvalho and Hall conjectured that the orbits forced by P0 are those ones that do not intersect a region Pw called pruning region, and that the forcing relation of periodic orbits depends basically of being able to determine it for homoclinic orbits. In this work we will conclude that the pruning method can be used for finding these pruning regions for certain infinite families of decorations w. This follows proving that after a finite number of applications of the differentiable pruning theorem one can eliminate all the bigons rel those orbits. Thus Pw is precisely the union of the pruning domains where the invariant manifolds were uncrossed.

Theorem 2. If w lies to one of following classes of decorations: maximal, P-list or star, then there S i exists a pruning region Pw such that CP w = Σ2 \ σ (Pw) up to topological finite-to-one semi- 0 i∈Z conjugacy which is injective on the set of non-boundary periodic points.

In section3 are defined all the concepts needed by theorem above. Thus the set of orbits that have to coexist with these homoclinic orbits is described completely. Among others results contained in the text, the pruning method allows us to prove a Milnor-Thurston-like forcing on maximal homoclinic orbits (Corollary 26):

0 0 0 Corollary 3. If w and w are maximal finite words satisfying that w >1 w and wb >1 wc then w w0 P0 >2 P0 , where >1 denotes the unimodal order and wb denotes the reverse word of w. A version of this result was proved by Holmes and Whitley [31] for homoclinic orbits that appear in strongly dissipative H´enonmaps. In section4 we introduce the pruning method, that is an algorithm which could allow us to find, given a homoclinic orbit P of an arbitrary Smale map f, another Smale map ψ without bigons rel P . Hence the basic piece of such map ψ has to contain all the dynamics forced by P . Unfortunately there is an inconvenient in our treatment: there is no guarantee that the method always stops in a finite number of steps; but even in that case there exists a pruning model, describing the dynamical core, which belongs to the isotopy class of a limit of hyperbolic pruning models [35]. 4 VALENT´IN MENDOZA

Now we would like to explain how the paper is organized. Section2 is devoted to relation between bigons and the forcing relation of homoclinic orbits. Section3 is devoted to the application of the differentiable pruning theory to the study of homoclinic horseshoe orbits and section4 will introduce the pruning method in a general form.

2. Dynamics forced by homoclinic orbits

Here we will define the notions that we are going to use. We assume that the reader has some familiarity with Smale maps and pseudoanosov homeomorphisms. Good references for these topics are [5] and [19].

2.1. Smale maps. Let f be a Smale map on the closed disk D2 and suppose that f has a unique non-trivial basic saddle set K which is a totally disconnected hyperbolic Cantor set. Suppose the domain of K, that is, an invariant open region containing K where the dynamics can be explained 2 by the symbol dynamics of K is ∆(K) = D \{s1, s2, ··· , sk} where si is a periodic point of f for all i = 1, ··· , k. See the precise definition of ∆(K) in [5]. Thus the non-wandering set of f is formed by K and a finite set of isolated saddles points, sinks and sources. We will suppose that f can be extended to ∂D2 as the identity f = Id. A saddle point x ∈ K is a s-boundary point if x is a boundary point of W u(x, f) ∩ K, that is, if x is an accumulation point only from one side by points in W u(x) ∩ K, or equivalently, if there exists a closed interval I ⊂ W u(x, f) having x as end-point such that Int(I) ∩ K = ∅. The set of s-boundary points is denoted by ∂sK. The u-boundary points are defined similarly; the set of the u-boundary points is denoted by ∂uK. It is known [38] that there exists a finite number of periodic saddle points ps, ··· , ps and pu, ··· , pu such that ∂ K = ∪ns W s(ps)
∩ K and ∂ K = ∪nu W u(pu)
∩ K. 1 ns 1 nu s i=1 i u i=1 i j We are going to study a homoclinic orbit P = {pj}j∈Z = {f (p0)}j∈Z included in the intersection of

Figure 1. A homoclinic orbit P = Orb(p0) to a fixed point p. the stable and unstable manifolds of a s- and u- boundary fixed point p, and let us suppose that the eigenvalues of Df(p) are positive. Figure1 shows an example. The orbit of a point x ∈ K by f will i be denoted by R = Orb(x) = {f (x)}i∈Z. We need the following definition. Definition 4. A bigon is a simply connected open region I bounded by a segment of a stable manifold s u θs ⊂ W (ps) and a segment of an unstable manifold θu ⊂ W (pu), where ps and pu are saddle points of K. DYNAMICS FORCED BY HOMOCLINIC ORBITS 5

(a) (b)

Figure 2. Wandering bigons.

There are two types of bigons. A bigon I is called wandering if it is disjoint from K. In this case s u ∂I = θs ∪ θu with θs ⊂ W (ps) and θu ⊂ W (pu), where ps and pu are boundary periodic points of K. Figure2 shows two wandering bigons. The second type of bigons is the following: A bigon I is called non-wandering if I ∩ K 6= ∅. In general, a non-wandering bigon contains a wandering bigon is its interior. If it is not the case, there are two possibilities: I contains a s-boundary periodic saddle point x whose free branch of W u(x) belongs to the basin of an attracting periodic orbit, or I contains a u-boundary saddle point x whose free stable manifold belongs to unstable set of a repelling periodic point. See figure3.

Figure 3. Non-wandering bigons.

If f has no bigons relative to a homoclinic orbit P then there are not non-wandering bigons and every bigon is as in figure2(a) where pj represents an element of the homoclinic orbit, that is, there exists a pj ∈ P such that {pj} ⊂ θs ∩ θu. In this case we say that K is exteriorly situated rel P . If P is a homoclinic orbit to a fixed point p, the following proposition proves that f restricted to K will be transitive providing that it does not have bigons.

Proposition 5. Let P be a homoclinic orbit to a fixed point p of a Smale map f on D2. If f does not have bigons rel P then f is transitive on its basic set K.

Proof. Suppose that there exist at least two transitive disjoint basic sets K1 and K2 such that {p}∪P ⊂ s u s u K1. If W (K1) ∩ W (K2) 6= ∅ then the elements pj of P have to be situated in W (K1) ∩ W (K2), −n because otherwise there would exist bigons rel to P . It is a contradiction since, in that case, lim f (pj) s u goes to p ∈ K1 ∩K2 when n goes to ∞, which is clearly a contradiction. Hence W (K1)∩W (K2) = ∅. u s Similarly we can prove that W (K1) ∩ W (K2) = ∅. Hence K1 and K2 are not homoclinically related. 2 Thus one can suppose that P ⊂ K1 and P ∩ K2 = ∅. Since D is simply connected, it follows that any homoclinic intersection happening in K2 creates wandering and non-wandering bigons rel to P . It is a contradiction with the hypothesis. Hence K2 = ∅.  6 VALENT´IN MENDOZA

Definition 6. A pair of periodic points x, y ∈ K is u-special if W u(x) ∪ W u(y) is accessible from inside boundary of a domain that is a continuous immersion of the open disk in D2 and belongs to D2 \ K. In an analogous manner, a pair of periodic points x, y ∈ K is s-special if W s(x) ∪ W s(y) is accessible from inside boundary of a domain that is a continuous immersion of the open disk in D2 and belongs to D2 \ K.

Figure 4. In (a) is pictured a pair of u-special points and in (b) is pictured a pair of s-special points.

See figure4 for examples.

Proposition 7. Let K be an exteriorly situated basic set of a Smale map f on MP . Then f is isotopic to a Smale map f ∗ on a basic set K∗ by a semiconjugacy τ : K → K∗ such that (1) τ(K) = K∗, τ ◦ f = f ∗ ◦ τ; (2) the set Z ⊂ K∗ of points z whose preimage τ −1(z) contains more than one point is such that −1 u u S s s u h (Z) = K ∩ (∪W (pi ) ∪W (pj )) where {pi } is the finite set of u-special periodic points s and {pj } is the finite set of the s-special periodic points; (3) K∗ does not have special pairs of points.

k Proof. Note that there exist a finite number of pair of periodic special points {xi, yi}i=1. Suppose that x and y be a u-special pair of boundary periodic points and let B be the region between W u(x) and u u u W (y). Then zipping B by a semi-conjugacy τ which identifies points r1 ∈ W (x) and r2 ∈ W (y), that can be joined by a stable leaf included in B, one obtains a Smale map isotopic to f whose dynamics is semiconjugated to f|K and has k − 1 special pairs of points. Applying the same process to this new map one can construct a Smale map with k − 2 special pairs of points. By induction we find a Smale ∗ map f satisfying the mentioned conclusions. 

2.2. The dynamics forced by a homoclinic orbit. In this section we will study a forcing relation on homoclinic orbits to a fixed point. As we have said in the introduction, in this case one must work with the Handel-Miller theory which is a generalization of the Nielsen-Thurston theory applied to end periodic homeomorphisms [29]. j Let P = {pj : pj = f (p0), ∀j ∈ Z} be a homoclinic orbit to a fixed point p of a Smale map f on 2 2 D . As said before let MP = Int(D ) \ P and let fP be the restriction of f to MP . Next lemma proves that fP has the topological type of an end-periodic automorphism of the noncompact surface with one attracting end and one repelling end. DYNAMICS FORCED BY HOMOCLINIC ORBITS 7

Lemma 8. The isotopy class of fP contains a map which is conjugated to an end-periodic homeo- morphism f on the non-compact hyperbolic surface S = R2 \ (Z × {0}). The map f has one attracting end and one repelling end.

Proof. In fact, if splitting open the stable manifold of p by a stable DA-isotopy (which was introduced in [42] and is the inverse process of zipping) one obtain a map f0 for which p is an attracting point and such that f0 has two saddles points x1 and x2. See figure5 for an example with the horseshoe orbit ∞01010110∞ homoclinic to the point p = 0∞. The stable DA isotopy transforms P into a heteroclinic n −n orbit Q = {qj}, that is, limn→+∞ f0 (qi) = x1 and limn→+∞ f0 (qi) = x2. See figure5(b). Then

(a) (b) (c)

Figure 5. DA-isotopies of f and their action around the fixed point p. making an unstable DA isotopy of f0 one get a Smale map f1 for which x2 is now an repelling point.

Furthermore f1 has two saddle fixed points denote by y1 and y2. See figure5(c). Thus Q is now n transformed in a heteroclinic orbit, that we will also called Q, satisfying that limn→+∞ f1 (qi) = x1 −n 2 and limn→+∞ f1 (qi) = y2. By a new isotopy one can push x1 and p to a point t1 in ∂D , and one 2 can push y2 and x2 to point t2 ∈ ∂D , to obtain a map f on the disk which has an orbit Q such n −n 2 2 that limn→+∞ f1 (qi) = t1 and limn→+∞ f1 (qi) = t2. Identifying Int(D ) with R and conjugating so that Q is identified with Z × {0}, we obtain a map also called f that has a basic compact basic set and two ends. The surface S = R2 \ (Z × {0}) is hyperbolic as was shown in [28, Section 3]. See example 15.  Example 9. In figure6 we have pictured the invariant manifolds of the map f, given by lemma8, ∞ ∞ for the horseshoe homoclinic orbit p0 = 010 · 10110 which has been represented in figure5(a). See section 3.1 for the description of the Smale horseshoe map.

Figure 6. Invariant manifolds of the end periodic Smale diffeomorphism f con- structed applying lemma8 to the horseshoe homoclinic orbit ∞010 · 10110∞. 8 VALENT´IN MENDOZA

We will follow the presentation of the Handel-Miller theory given by Cantwell and Conlon in [10] for end-periodic homeomorphisms, in particular, by lemma8, we can suppose that fP is defined on S, has a basic set K and P = Z × {0}. Let Se be the universal cover of S which is identified with the 2 Poincar´edisk D , π : Se → S be the recovering map, fe : Se → Se be a fixed lift of f and fb : S∞ → S∞ its canonical extension to the closed unit disk S∞ := D ∪ ∂D. First Cantwell and Conlon have defined 1 the Handel-Miller map h : S → S isotopic to f such that h = f in the boundary S∞ = ∂D. Then they extended it using a very interesting construction and were able to find two bi-laminations Γ± and Λ± on S which are invariant by h. It follows that Λ± is formed by immersed geodesics which are complete 1 and whose end-points of their lifts are in S∞. Denotes the support of Λ± by |Λ±|.

Definition 10. The dynamical core C of f is the intersection of the invariant laminations |Λ+| ∩ |Λ−| of the Handel-Miller map h associated to f.

The homeomorphism h is not uniquely determined but it is unique when it is restricted to the core.

Theorem 11. [10, Corollary 10.13] The dynamical core C of f is uniquely determined up to a topo- logical conjugacy.

The dynamical core relative to a homoclinic orbit P , denoted CP , is the dynamical core of f on S. Since f and g on S are isotopic if and only there are lifts to Se such that fb and gb agree on the ideal boundary [9] it follows that CP is a topological invariant under ambient isotopies. So CP is persistent which means that every g ambiently isotopic to f on S has the same core. Thus one can give the following definition of forcing in this context.

Definition 12. We will say that the dynamics forced by P is the dynamics of h : CP → CP , that is, an h-orbit Q is forced by P , denoted P >2 Q, if Q ⊂ CP . Note that this definition involves periodic and non-periodic orbits, indeed, it follows from [10, theorem 9.2] that h|CP is conjugated to a two-ended Markov shift of finite type.

If a Smale map f does not have bigons relative to P , theorem 14 below proves that CP agrees with its basic set K up to topological finite-to-one semi-conjugacy. It depends of the following result of Grines [23], extended to a surface of infinite type, who has proved analogous conclusions for hyperbolic attractors on surfaces of finite type.

Theorem 13. [23] Let f be a Smale map on D2 with a basic set K, and let P be a homoclinic orbit to a fixed point p ∈ K. If K is exteriorly situated relative to P without special points then there exist two u s u s u s geodesics laminations (L , L ), a map f0 on L ∩ L and a continuous map ι : L ∩ L → K which is homotopic to the identity such that u s (1) ι(L ∩ L ) = K, f ◦ ι|Lu∩Ls = ι ◦ f0|Lu∩Ls ; (2) the set B ⊂ K of points b whose preimage ι−1(b) contains more than one point is formed by s u u u s K∩(∪W (pi )) where {pi } is the finite set of the u-boundary periodic points and K∩(∪W (pi )) s where {pi } is the finite set of the s-boundary periodic points; (3) the set ι−1(b) consists of exactly two points belonging to distint boundary geodesics of Lu or Ls.

Proof. It is sufficient to prove the conclusions for the map f : S → S given in lemma8. Hence we can suppose that f is an end periodic map with a basic saddle set K and two ends, defined on the end periodic surface S. Using the construction of Grines [23], for each x ∈ K we have that DYNAMICS FORCED BY HOMOCLINIC ORBITS 9

s Figure 7. The construction of the geodesics lx.

• if both components of W s(x) \{x} intersect K then W s(x) lifts in the Poincar´edisk Se = D2 s 1 + − to a curve Wf (xe) which has two-endpoints at S∞, ν and ν . Taking the geodesic tightening of Wfs(x), we construct a geodesic els , that is, the geodesic passing trough ν+ and ν−. See e xe figure7(a).

We set Les to the set of {els : x is a non-boundary periodic point} and Ls = π(Les). Denote Ls to the xe closure of Ls on S. Note that if x is an u-boundary periodic point but not s-boundary then, taking u s,1 s,2 s limits of geodesics from both sides of W (x), there exists two geodesics lx and lx in L , as in figure 7(b), that can be considered the geodesic tightening of W s(x). See [24, Chapter 9] for a detailed explanation. u s u Analogously we state the geodesic lamination L . Defining the maps ι and f0 by ι(lx ∩ lx) = x and s u s u f0(lx ∩ lx) = lf(x) ∩ lf(x) as in [1], one can prove that s u s u ι ◦ f0(lx ∩ lx) = f(x) = f ◦ ι(lx ∩ lx).

 Now we will prove the main result of this section.

Theorem 14. Let f be a Smale map on D2 with a basic set K, and let P be a homoclinic orbit to a fixed point p ∈ K. If K is exteriorly situated relative to P without special points then CP = K up a topological finite-to-one semi-conjugacy ι : CP → K which is injective on the set of non-boundary points.

u s u s Proof of Theorem 14. We will prove that Λ+ = L and Λ− = L , that is, CP = L ∩ L . It is sufficient to prove that (W u(K),W s(K)) is a bilamination satisfying the axioms 1–4 given in [10, Section 10]. Then by the isotopy theorem [10, Theorem 10.14], (Lu, Ls) = (g(W u(K)), g(W s(K))) is the Handel- Miller lamination, where g(λ) = λg is the geodesic tightening of λ for any unstable or stable leaf as it was defined in the previous paragraph. Thus K = CP up to an ambient isotopy and a finite-to-one semiconjugacy. (i) We will follow the same argument of the proof of [33, Theorem 5.3] by Lewowicz and Ures.

Take a simply closed curve γ = γs ∪ γu which is bounded by a stable segment and an unstable n segment. Now let x be a non-boundary fixed point of f which belongs to K and let xe be u one of its lifts. Since K does not have bigons then W (xe, fe) intersects every lift of γs at most u u a point. Since W (x, f) intersects infinitely many times γs then W (xe, fe) has exactly two 10 VALENT´IN MENDOZA

1 u s end-points at the boundary S∞ of the universal cover of M. It proves that (W (K),W (K)) satisfies Axioms 1 and 2.

(ii) By lemma8 there exist neighbourhoods U+ and U− of the ends t1 and t2, such that the sets u s J+ = ∂U+ and J− = ∂U− satisfy: (1) W (K) ∪ J− and W (K) ∪ J+ are each sets of disjoint u s pseudo-geodesics, (2) W (K) is transverse to J+ and W (K) is transverse to J−, and (3) no u s leaf of W (K) can meet J+ so as to form bigon and no leaf of W (K) can meet J− so as to form a bigon. See example 15. Then (W u(K),W s(K)) satisfies Axiom 4. s (iii) We will prove that, for any γs ⊂ W (K), γs is included in the closure of the set of non-escaping s component of the positive juncture J+. Suppose that γs = W (x) where x is n-periodic. Then, u u by item (ii), W (x) intersects J+. Thus there exists a segment γu ⊂ W (x) with one end-point −in in z ∈ J+ and the other in x. Since limi→+∞ diam(f (γu)) = 0, by the Lambda lemma,

given an open set U containing a compact segment of γs, a segment α ⊂ J+ containing z and −in  > 0, there exists an i0 such that for all i ≥ i0, f (α) ∩ U is -close to γs ∩ U. Hence γs is

at the closure of the non-escaping components of J+. Since periodic points are dense in K, it s follows that, for all x ∈ K, γs = W (x) is at the closure of the non-escaping components of u s J+ as well. Thus (W (K),W (K)) satisfies Axiom 3. 

Thus theorem 14 can be compared with a result by Lewowicz and Ures [33] that proves that the basic set of a Smale map on a surface of finite topology is included in the persistent set given by Handel [27], if K is exteriorly situated, that is, if there are no bigons.

∞ Example 15. Consider the homoclinic orbit Θ which corresponds to the horseshoe orbit p0 = 010 · 10∞ which is represented in figure8(a). Note that F does not have bigons relative to Θ then, applying theorem 14, the dynamics forced by this orbit is the full horseshoe up to a finite-to-one- semiconjugacy. Note that f has an infinite orbit of a 1-pronged singularity at the elements of Θ. Applying the zero- entropy equivalence relation, stated in [15], we obtain the tight horseshoe defined on S2 which is a generalized pseudo-Anosov map φ having an infinite orbit of 1-pronged singularities. See figure8(b).

Figure 8. The tight horseshoe in the sphere S2.

The laminations of the end periodic map associated to this homoclinic orbit are given in figure9. −2 In this case the map f of lemma8 lies to the isotopy class of σi τ where σi is a map that interchanges the points pi and pi+1 and τ is the translation end map. DYNAMICS FORCED BY HOMOCLINIC ORBITS 11

Figure 9. The laminations of the end periodic map associated to the horseshoe homoclinic orbit ∞010 · 10∞.

2.3. Pruning theory. Pruning is a technique introduced by A. de Carvalho [14] for eliminating orbits of a homeomorphism in a controlled manner, that is, for destroying dynamics contained in the interior of simply connected closed regions that we can define dynamically. Here we will use the differentiable version of pruning that the author, in a joint work with A. de Carvalho, has developed in the forthcoming paper Differentiable pruning and the hyperbolic pruning front conjecture [18]. The main ideas are the following. Let f be an orientation-preserving Smale diffeomorphism on a surface S with a basic set K. Let D s u be a simply connected domain bounded by two segments θs and θu with θs ⊂ W (ps) and θu ⊂ W (pu) where ps and pu are saddle periodic points in K with periods ns and nu, respectively.

Definition 16. The domain D is a pruning domain if its boundary satisfies the following properties n (i) f (θs) ∩ Int(D) = ∅, for all n ≥ 1. −n (ii) f (θu) ∩ Int(D) = ∅, for all n ≥ 1.

An easy consequence of definition 16 is next lemma:

Lemma 17. Let D be a pruning domain. Then the curves θs and θu satisfy: n ˚ (i) Either ps ∈ θs or F (θs) ∩ θs = ∅, for all n > 0. −n ˚ (ii) Either pu ∈ θu or F (θu) ∩ θu = ∅, for all n > 0.

The differentiable version of the pruning theorem is the following result.

Theorem 18 (Differentiable Pruning Theorem). If D is a pruning domain for f then there exists a diffeomorphism ψ, isotopic to f, satisfying the following properties: (i) ψ is a Smale map for which every point of int(D) is wandering for ψ,;

(ii) the non-wandering set of ψ consists of a saddle set Kψ and a finite set of isolated saddle points, sinks and sources; 0 (iii) ψ, restricted to Kψ, is semiconjugated to f, restricted to a subset K ⊂ K by a (at most 4-to-1) 0 semiconjugacy ζ : Kψ → K satisfying [ K0 = {x ∈ K : Orb(x, f) ∩ Int(D) = ∅} = K \ f i(Int(D)); and i∈Z (iv) ζ is injective on the set of non-boundary periodic points.

Sketch of the proof of theorem 18. The ideia for proving theorem 18 is to substitute f by a Smale map g1 which can be pruned. It is done, if necessary, constructing DA-maps diffeotopic to f which are difeomorphisms similar to those ones constructed by Williams [42] for Anosov diffeomorphisms. Thus

• we have to splitting open the stable manifold of ps by a stable DA isotopy to obtain a map g0

for which ps is an attracting periodic point, 12 VALENT´IN MENDOZA

u • then to splitting open the unstable manifold of pu by g0, W (g0, pu), by a unstable DA isotopy

to obtain a Smale map g1 isotopic to f for which pu is a repelling periodic point.

The construction of such attracting and repelling DA isotopies, around ps and pu respectively, were defined in [4, Section 7] and [21, Section 2.2.2] in a general setting using DA-bifurcations. Thus g1 has a ns-periodic basin of attraction BA(ps) and a nu-periodic basin of repulsion BU(pu). See figures 10(a) and 10(b). Moreover g1 has a hyperbolic basic set K1, and there exists a semi-conjugacy ζ0 : K1 → K which is at most 4-to-1 such that

ζ0 ◦ g1(x) = f ◦ ζ0(x), ∀x ∈ K1.

Figure 10. A pruning difeotopy.

Next we define a difeotopy St of the identity with support in Int(D) such that S1 takes a curve

αu ⊂ BU(pu) ∩ Int(D) to a curve in αs ⊂ BA(ps) ∩ Int(D), S(αu) = αs, in such a way that:

• S1(D \ BU(pu)) ⊂ BA(ps); −1 • S (D \ BA(ps)) ⊂ BU(pu).

See figures 10(c) and 10(d). Then composing St with g1 one get a diffeotopy ft = St ◦ g1 of f. Thus ft takes the non-wandering dynamics of g1 in Int(D) and pushes it to the attracting basin of ps. So the pruning diffeomorphism associated to D will be given by ψ := f1, the end of the diffeotopy. That ψ is a Smale map with a basic set Kψ included in K1. Thus the semi-conjugacy is given by ζ := ζ0|Kψ .  The following proposition proves that the topology of the invariant manifolds of ψ within D is known: they are just deformations by S1 of the invariant manifolds of f.

Proposition 19. Let γs ⊂ D and γu ⊂ D be segments of stable and unstable manifolds of a point x ∈ Kψ by ψ, respectively. Then 0 u 0 (a) If ps ∈/ θs, there exists a segment γu ⊂ W (x, g1) such that γu = S1(γu ∩ D). DYNAMICS FORCED BY HOMOCLINIC ORBITS 13

0 u (b) If ps ∈ θs, there exist a non-negative integer n0 and a segment γu ⊂ W (x, g1) such that n0 0 γu = ψ ◦ S1(γu ∩ D). 0 s 0 (c) If pu ∈/ θu then there exists a segment γs ⊂ W (x, g1) such that γs = γs. 0 s (d) If pu ∈ θu then there exist a non-negative integer n0 ≥ 1 and a segment γs ⊂ W (x, g1) such −n0 0 that γs = ψ (γs).

Proof. By lemma 17,

n ns (p1) either ps ∈/ θs and g1 (BA(ps) ∩ Int(D)) ∩ Int(D) = ∅, ∀n ≥ 1, or g1 (BA(ps) ∩ Int(D)) ⊂ BA(ps) ∩ Int(D); and −n nu (p2) either pu ∈/ θu and g1 (BU(pu) ∩ Int(D)) ∩ Int(D) = ∅, ∀n ≥ 1 or g1 (BU(pu) ∩ Int(D)) ⊂ BU(pu) ∩ Int(D). Let z ∈ W u(x, ψ) ∩ Int(D) and let n ≥ 1 be the smallest integer such that ψ−n(z) ∈ Int(D) then −i −i −1 ψ (z) = g1 ◦ S1 (z), ∀i = 0, ..., n. Since Int(D) \ BA(ps) is in the basin of an attractor point for −1 −n −1 −1 n ψ , it follows that g1 ◦ S1 (z) ∈ BA(ps) ∩ Int(D). Hence S1 (z) ∈ g1 (BA(ps) ∩ Int(D)). n n n By (p1), if ps ∈/ θs then z ∈ S1(g1 (BA(ps) ∩ Int(D))) = g1 (BA(ps) ∩ Int(D)). So z ∈ g1 (BA(ps) ∩ −i −i −1 Int(D)) ∩ Int(D) = ∅. It is a contradiction. So ψ (z) = g1 ◦ S1 (z), ∀i ≥ 1. This implies that −1 u u S1 (z) ∈ W (x, g1), or z ∈ S1(W (x, g1)). This proves (a). −n0 Now suppose that ps ∈ θs. Let n0 be the biggest non-negative integer such that ψ (z) ∈ Int(D).

−j −n0 −j −1 −n0 −1 −n0 u n0 So ψ (ψ (z)) = g1 ◦ S1 (ψ (z)), ∀j ≥ 1. Then S1 (ψ (z)) ∈ W (x, g1). Then z ∈ ψ ◦ u S1(W (x, g1)). This proves (b).

Figure 11. Effects of the pruning isotopy on the invariant manifolds of f. The invariant manifolds showed correspond to the pruning diffeomorphism ψ.

s n −1 Let z ∈ W (x, ψ)∩Int(D) and let n ≥ 0 be the smallest integer such that ψ (z) ∈ g1 (Int(D)). That i i −1 is ψ (z) = g1(z), ∀i = 1, .., n. It follows from definition of S1 that g1 (Int(D) \ BU(pu)) is included n −1 in the basin of one attractor point of ψ, so g1 (z) ∈ g1 (BU(pu) ∩ Int(D)). If pu ∈/ θu, by (p2), 14 VALENT´IN MENDOZA

−n −n−1 g1 (BU(pu) ∩ Int(D)) = ∅, ∀n ≥ 1. In this case, we have a contradiction since z ∈ g1 (BU(pu) ∩ i i s Int(D)) ∩ Int(D). So ψ (z) = g1(z), ∀i ≥ 1. This implies that z ∈ W (x, g1). It proves (c). n0 Now suppose that pu ∈ θu then let n0 be the biggest non-negative integer such that ψ (z) ∈ Int(D).

j n0 j n0 n0 s −n0 u So ψ ◦ψ (z) = g1 ◦ψ (z), ∀j ≥ 1. Then ψ (z) ∈ W (x, g1). This implies that z ∈ ψ (W (x, g1)). So we have proved (d). 

Proposition 19 can be useful in applications where it is important to know how are modified the

ns nu stable and unstable manifolds under the pruning isotopy. If the derivatives Df (ps) and Df (pu) have positive eigenvalues, then the invariant manifolds of ψ within D are pictured in figure 11 for some cases considered in proposition 19: (i) if ps ∈/ θs and pu ∈/ θu, (ii) if pu ∈ θs and pu ∈/ θu, (iii) if ps ∈/ θs and pu ∈ θu, and (iv) if ps ∈ θs and pu ∈ θu. The reader is encouraged to draw the invariant manifolds of ψ if one of those derivatives has a negative eigenvalue. From proposition above, for knowing how the bigons are created or destroyed we just have to concern with the deformation of the invariant manifolds by S1 within D. So ψ must just be understood as f without intersections of stable and unstable manifolds within Int(D) (and then within all its iterates).

3. Forcing on homoclinic horseshoe orbits

In this section we will study certain homoclinic orbits of the Smale horseshoe, one of the most famous diffeomorphism in dynamical systems. We will determine the dynamical core forced by them exhibiting the sequence of pruning maps (or pruning domains) that are sufficient for eliminating all the bigons relative to these orbits. w ∞ ∞ 0 0 ∞ We are only concerned with homoclinic orbits P = P0 to 0 which have as code 011w110 where w is a finite word called the decoration of P (See section below). By a Handel’s result, cited in [8], it is known that the homoclinic orbit ∞010 · 10∞ forces every horseshoe orbit as it was shown in example 15. So our study here is devoted to try of making similar conclusions for other homoclinic orbits, that is, which are the orbits (periodic or not) forced by a given homoclinic one.

There exist some methods for determining CP for homoclinic orbits. The first one, introduced in Hulme’s thesis [30], is a generalization of the Bestvina-Handel algorithm to construct pseudo- Anosov-like representatives of certain homoclinic or heteroclinic braids which can be considered as translation-ends classes. The second one was stated by Collins in [11] where the trellis of a homoclinic or heteroclinic orbit allows to construct a graph representative which contains the dynamics forced by that orbit and, in certain cases, also allows to define a hyperbolic map realising the entropy bound [12]. There are two main differences between these methods and our technique. The first one is the fact that pruning theory can be applied wherever the Collins’s method does not work. For example, in ∞ 00 ∞ [12, Example 4.3] it is pointed out that for the trellis associated to the orbit P = 011110 there is no a minimiser hyperbolic diffeomorphism. As we will see in section 3.4, P corresponds to the star orbit 1/3 P0 for which there exists a well-defined pruning region, and so, a Smale map realising CP . Other difference consists in that, once we have found a pruning region for an individual homoclinic orbit, it is easy to generalize that region for an infinite family of decorations; the reason for it is the observation that the pruning regions seem to depend only on the combinatorics of the orbit and do not depend on the coordinates themselves. It will be interesting to study the relation between the combinatorics and the pruning regions in a general context. DYNAMICS FORCED BY HOMOCLINIC ORBITS 15

3.1. Smale horseshoe. The Smale horseshoe is a well-known hyperbolic diffeomorphism F on D2 which acts as in figure 12(a). Its non-wandering dynamics consists of an attracting fixed point in the left semi-circle, and a compact basic set K included in the regions V0 and V1. Furthermore, there exists a conjugacy between F , restricted to K, and the shift σ, defined on the compact set Σ2 = {0, 1}Z. i Thus each point x ∈ K is represented by a sequence s = (si)i∈Z where si = 0, if F (x) ∈ V0, and i si = 1, if F (x) ∈ V1.

Figure 12. The Smale horseshoe map and its homoclinic tangle.

In the following sections, a point x ∈ K will be always represented by its symbol code. Each point x ∈ K has two invariant manifolds W s(x) and W u(x) that are dense in K and it will be supposed that they are vertical and horizontal, respectively. If x = s− · s+, where s+ and s− are unilateral + N sequences in the compact space Σ2 = {0, 1} , then we can project x along its invariant manifolds to ∞ ∞ ∞ the lowest unstable manifold of 0 for obtaining 0 · s+, and to the leftmost stable manifold of 0 ∞ + for obtaining s− · 0 . One can compare two points of K using their positions s+ and s− in Σ2 given by the unimodal order >1 which relates two symbol sequences by the following rule: If s = s0s1 ··· + and t = t0t1 ··· are two sequences in Σ2 with si = ti for all i ≤ k and sk+1 6= tk+1, then s >1 t when: Pk (O1) i=0 si is even and sk+1 > tk+1, or Pk (O2) i=0 si is odd and sk+1 < tk+1. So s >1 t if s = t or s >1 t. Moreover let s = s− · s+, t = t− · t+ be points of Σ2 then • if s+ >1 t+, we say that s >x t, and • if s− >1 t−, we say that s >y t.

Denote by w to the infinite repetition ww... of a word w. We will say that a word w = w1w2 ··· wM PM is even if i=1 wi is even; otherwise w is odd. Let wb = wM wM−1 ··· w1 be the reversal word of w. Here an horseshoe orbit will be denote by R. If R is a periodic orbit then the code of R, denoted by cR is the symbolic representation of the rightmost point of R in the unimodal order. When R is not periodic, the code of R can be taken as the symbolic representation of some of its points.

Note that the Smale horseshoe has only one bigon I0 (and its iterates) formed by a stable segment 0 0 ∞ ∞ ∞ ∞ θs and an unstable segment θu which intersect at the homoclinic points 011 · 10 and 010 · 10 . See figure 12(b). Observe that F does not have special pairs of points.

3.2. Maximal decorations. In this section we will study maximal decorations. A decoration w i is maximal if w0 and w1 are maximal codes in the unimodal order, that is, σ (w0) 61 w0 and i σ (w1) 61 w1, for all i > 1. 16 VALENT´IN MENDOZA

Example 20. Consider the decoration w = 10. Since w0 = 100 and w1 = 101 are maximal codes, 10 one get that w is maximal. Among the four orbits within the isotopy class of P0 , choose the orbit P = ∞010 · w110∞ = ∞010 · 10110∞. Note that there exists a maximal region D, containing the bigon

Figure 13. The maximal region D is not a pruning domain, but it can be reduced to a pruning domain D1.

∞ ∞ ∞ ∞ I, bounded by a stable segment γs of 010 · 10110 and an unstable segment γu of 010101 · 10 . 3 Since F (γs) ∩ Int(D) 6= ∅, the domain D is not a pruning domain. See figure 13(a). However one can decrease γs to a segment θs ⊂ γs such that θs and a segment θu bound a pruning domain D1. It is ∞ ∞ not difficult to see that θu is included in the unstable manifold of (w0) = (100) . See figure 13(b).

Let ψ be the pruning map associated to D1. As one will see below, D1 is sufficient for eliminating all 10 10 the bigons relative to P0 , that is, for proving that ψ does not have bigons relative to P0 . Figure 14

10 Figure 14. A representation of the dynamical core of P0 in the symbol plane and the invariant manifolds of ψ. shows the orbits (of periods less than 16) that do not intersect D1 which are the orbits that belong to 10 the dynamical core forced by P0 and the invariant manifolds of ψ. It was pictured with the program Domains.exe available in [36] which is a program to prune the Smale horseshoe using at most 5 pruning domains.

A similar argument to this one used in example above works for an arbitrary maximal decoration. DYNAMICS FORCED BY HOMOCLINIC ORBITS 17

Definition 21. Let w be a maximal decoration. We have to distinguish two cases as follows:

(a) If w is even then define Pw = Int(D1) where D1 is the pruning domain bounded by a stable s ∞ ∞ u ∞ segment θs ⊂ W ( 010 · w010 ) and an unstable piece θu ⊂ W ((w1) ) whose end-points are the heteroclinic points ∞(w1) · w010∞ and ∞(w1)w0 · w010∞.

(b) If w is odd then define Pw = Int(D1) where D1 is the pruning domain bounded by a stable s ∞ ∞ u ∞ segment θs ⊂ W ( 010 · w110 ) and an unstable piece θu ⊂ W ((w0) ) whose end-points are the heteroclinic points ∞(w0) · w110∞ and ∞(w0)w1 · w110∞.

These sets are represented in the symbol plane in figure 15 and are called pruning regions associated to w. The horizontal lines represent the unstable manifolds and the vertical lines represent the stable

(a) (b)

Figure 15. The set Pw when w is even (a) and odd (b).

w manifolds. So the following theorem describes the dynamical core forced by P0 . Theorem 22. Let w be a maximal decoration. Then

[ i C w = {x ∈ Σ : Orb(x,F ) ∩ P = ∅} = Σ \ σ (P ) P0 2 w 2 w i∈Z up to a finite-to-one semiconjugacy which is injective on the set of non boundary periodic points.

The following lemma is needed for proving theorem 22.

Lemma 23. Let w = w1 ··· wM be a maximal decoration. Then (i) If w is even then, for all i = 1, ··· ,M,

∞ ∞ i ∞ ∞ wiwi+1 ··· wM 010 61 w010 and σ ((w1) ) 61 w010 . (ii) If w is odd then, for all i = 1, ··· ,M,

∞ ∞ i ∞ ∞ wiwi+1 ··· wM 110 61 w110 and σ ((w0) ) 61 w110 .

Proof. Let us only to prove item (i). If there exists k < M − i such that wi ··· wi+k = w1 ··· w1+k and wi+k+1 6= wk+1, then both inequalities in (i) hold trivially. Suppose that wi ··· wM = w1 ··· wM−i+1. We have two cases: 0 ∞ ∞ • if wi ··· wM is even, since (wi ··· wM 1w1 ··· wi−1) 61 (w0) , it follows that wM−i+2 = ∞ ∞ 1. Thus wi ··· wM 0 61 w which implies that wi ··· wM 010 61 w010 . Suppose that i ∞ ∞ ∞ σ ((w1) ) = (a1 ··· aM aM+1) 61 (w1) . If a1 ··· aM = w1 ··· wM then aM+1 = 0, which implies that σi((w1)∞) = (w0)∞. It is contradiction because (w0)∞ is maximal. Thus i ∞ ∞ a1 ··· aM and w1 ··· wM disagree and then a1 ··· aM 61 w. Hence σ ((w1) ) 61 w010 . 18 VALENT´IN MENDOZA

• if wi ··· wM is odd the proof follows the same arguments. 

w ∞ ∞ Proof of theorem 22. Let P0 be the homoclinic orbit 010w0 · 10 where w is maximal and even with length M. Consider the domain D1 as in definition 21 which clearly contains I0. Note that ∞ ∞ D1 is bounded by two segments θs and θu where θs contains 010 · w010 and θu contains the ∞ ∞ ∞ periodic point pu = (w1) . So they are defined by the heteroclinic points A1 = (w1) · w010 and ∞ ∞ s ∞ u ∞ A2 = (w1)w0 · w010 , that is, θs ⊂ W (0 ) and θu ⊂ W ((w1) ).

Figure 16. The invariant manifolds of F .

i ∞ (m1) By lemma 23(i), it follows that F (θs) is to the left of w010 for all i = 1, ··· ,M − 1. Then i F (θs) ∩ Int(D1) = ∅ for i = 1 ··· M − 1; M ∞ ∞ M (m2) since F (A1) = (w1)w · 010 , it follows that F (θs) ∩ Int(D1) = ∅; M+1 ∞ ∞ M+1 (m3) since F (A1) = (w1)w0·10 , it follows that F (A1) belongs to the horizontal unstable M+1 leaf that contains A2, and thus F (θs) ∩ Int(D1) = ∅. i ∞ ∞ i (m4) since F (A1) = (w1)w01 ··· 0·0 , ∀i > M +1, it follows that F (θs)∩Int(D1) = ∅, ∀i > M +1. i n ˚ Thus for all i > 1, F (θs) ∩ Int(D1) = ∅. Hence F (θs) ∩ θu = ∅ for all n ≥ 1 which implies that −n ˚ −n −n F (θu) ∩ θs = ∅ for all n ≥ 1. So if for some n ≥ 1, F (θu) ∩ Int(D1) 6= ∅ then F (θu) ⊂ Int(D1). ∞ ∞ This is not possible since (w1) ∈ θu and, by lemma 23(i), Orb((w1) ) ∩ Int(D1) = ∅. Then D1 is a M+1 pruning domain. Figure 16 shows D1 and its (M + 1)-iterate F (D1). By the differentiable pruning theorem 18, there exists a Smale map ψ, isotopic to F , with a basic set Kψ which is semiconjugated to the set

[ i (1) Σ2 \ σ (Pw) = {x ∈ Σ2 : Orb(x,F ) ∩ Pw = ∅} i∈Z by a semiconjugacy which is injective on the set of periodic orbits. As in the proof of theorem 18, pu is a repelling point of ψ and has an repelling basin BU(pu). See figure 18. Since the pruning isotopy is supported in D1 and the pruning map uncrosses the invariant manifolds inside D1 and its iterates 0 (by proposition 19), it follows that ψ has a bigon I which is a deformation of I0 and does not have 0 ∞ ∞ any bigon inside D1. The bigon I has the point 010 · w010 in its boundary. See Figure 18. DYNAMICS FORCED BY HOMOCLINIC ORBITS 19

A combinatorial analysis on the iterates of the pruning domain gives us the code of the boundary periodic points of ψ.

Proposition 24. The basic set Kψ of ψ has an u-boundary 3(M +1)-periodic orbit Bw whose unstable manifolds define a principal region of 3 sides that contains a (M + 1)-periodic singularity Sw. If w is even then the code of Bw is w1w0w1 and the code of Sw is w1, and if w is odd then the code of Bw is w0w1w0 and the code of Sw is w0.

Proof. Suppose that w is even. Iterating (M + 1) times the domain D1, it follows that the invariant 1 manifolds are uncrossed in the region C ⊃ D1 defined by

1 ∞ ∞ ∞ ∞ C = {p : w010 6x p 6x 10 ,A2 6y p 6y (w1)w0w1 · w010 } ∞ which is between the unstable leaf of A2 and the unstable leaf l1 passing through the point (w1)w0w1· ∞ w010 . See figure 17(a). Iterating M +1 times the leaf l1 it follows that the uncrossing happen between

(a) (b)

(c) (d)

Figure 17.

∞ ∞ ∞ l1 and the unstable leaf l2 passing through the points (w1)w0w1w0 · w010 and (w1)w0w1w1 · ∞ w010 . Iterating M + 1 times the leaf l2 one can conclude that the invariant manifolds are uncrossed 2 1 in the region C ⊃ C ⊃ D1 defined by

2 ∞ ∞ ∞ ∞ ∞ ∞ C = {p : w010 6x p 6x 10 , (w1)w0w1w1w0 · w010 6y p 6y (w1)w0w1 · w010 } 20 VALENT´IN MENDOZA

∞ ∞ which is between the leaves l1 and l3 that contains the point (w1)w0w1w1w0 · w010 . See figure 17(b).

Repeating the process a second time using the leaves l3 and l1 one can show that the invariant manifolds are uncrossed in the region

∞ ∞ 3 w010 6x p 6x 10 , C = {p : ∞ 2 ∞ ∞ ∞ } (w1)w0(w1w1w0) · w010 6y p 6y (w1)w0(w1w1w0)w1 · w010 between unstable leaves passing through the points ∞(w1)w0(w1w1w0)2·w010∞ and ∞(w1)w0(w1w1w0)w1· w010∞. See figure 17(c). Repeating it n + 1 times one can prove that the invariant manifolds are uncrossed in the region

∞ ∞ n+1 w010 6x p 6x 10 , C = {p : ∞ n ∞ ∞ n−1 ∞ }. (w1)w0(w1w1w0) · w010 6y p 6y (w1)w0(w1w1w0) w1 · w010 In the limit the invariant manifolds are uncrossed in the region

∞ ∞ ∞ w010 6x p 6x 10 , C = {p : ∞ ∞ ∞ ∞ }. (w1w1w0) · w010 6y p 6y (w1w1w0)w1 · w010 between the unstable leaves passing through ∞(w1w1w0) · w010∞ and ∞(w1w1w0)w1 · w010∞ which contain the periodic points (w1w1w0)∞ and (w1w0w1)∞, respectively. Thus the orbit of (w1w1w0)∞ is an unstable boundary periodic orbit. See figure 17(d). 

(a) (b)

Figure 18. The invariant manifolds of ψ.

Let us to finish the proof of theorem 22 proving that ψ has no bigons. From proposition 24, the region

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ A1 = [ (w1w0w1) · w1w0w010 , (w1w0w1) · 10 , (w1w1w0) · 10 , (w1w1w0) · w1w0w010 ], which is the rectangle whose boundary is formed by segment of invariant manifolds, does not contains bigons. Using proposition 19 and the combinatorics of the edges of A1, one can see that the rectangle −3(M+1) ∞ ∞ ∞ ∞ ψ (A1) = [ (w1w0w1) · w1w0w1w1w0w010 , (w1w0w1) · w1w0w110 , ∞(w1w1w0) · w1w1w010∞, ∞(w1w1w0) · w1w1w0w1w0w010∞], DYNAMICS FORCED BY HOMOCLINIC ORBITS 21

−2(M+1) has a stable side included in the stable side of ψ (D). See figure 18(a). Since A1 has also a −2(M+1) stable side in ψ (D1), one can extend A1 to a bigger rectangle ∞ ∞ ∞ ∞ A2 = [ (w1w0w1) · w1w0w1w1w0w010 , (w1w0w1) · 10 , ∞(w1w1w0) · 10∞, ∞(w1w1w0) · w1w1w0w1w0w010∞] without bigons. See figure 18(b). Repeating this process for A2, we can construct a rectangular region

A3 ⊃ A2 such that ∞ 2 ∞ ∞ ∞ A3 = [ (w1w0w1) · (w1w0w1) w1w0w010 , (w1w0w1) · 10 , ∞(w1w1w0) · 10∞, ∞(w1w1w0) · (w1w1w0)2w1w0w010∞].

Repeating this process ad infinitum one obtains a region

∞ ∞ ∞ ∞ A∞ = [ (w1w0w1) · (w1w0w1) , (w1w0w1) · 10 , ∞(w1w1w0) · 10∞, ∞(w1w1w0) · (w1w1w0)∞] without bigons. It is clear that A∞ is periodic of period 3(M + 1) and then there are no bigons −(M+1) −2(M+1) in the region A∞ ∪ ψ (A∞) ∪ ψ (A∞). Thus one can conclude that ψ does not have w bigons relative to P0 . By theorem 14 and (1), the orbits which do not intersect Int(D1) belong to the w dynamics forced by P0 up a finite-to-one semiconjugacy injective on the set of non-boundary periodic points.

By proposition 19, the unstable invariant manifolds of ψ1 are just deformations of unstable invariant 0 manifolds of F and so I0 is pushed by the pruning isotopy to I . Using proposition 24, the crossings ∞ ∞ ∞ between invariant manifolds situated in [A1,B1]s are situated in the segment [A1 ,B1 ]s where A1 ∞ u ∞ and B1 are points at W ((w0w1w1) ). See figure 18(b). Iterating (M + 1) times, one see that there M+1 ∞ ∞ are no bigons in ψ ([A1 ,B1 ]s) and so there ψ does not have bigons. If w is odd the proof follows the same lines with minor modifications. 

Figure 19 shows how proposition 24 works when w is even: three points with codes w1w0w1, w0w1w1 e w1w1w0 define a ideal polygon of 3 sides that contains the point (w1)∞, creating a 3- pronged singularity.

Figure 19. If w is even, the orbit of the unstable boundary point with code w1w0w1 can be collapsed to the periodic orbit with code w1. 22 VALENT´IN MENDOZA

10 ∞ ∞ Example 25. Consider the homoclinic orbit P0 of the point p0 = 010 · 10110 with maximal decoration of example 20, and let ψ be its hyperbolic pruning map associated which, by proposition 24, has a period 3 3-pronged singularity at the periodic orbit 100.

w From theorem 22 it is easy to determine the forcing relation between two homoclinic orbits P0 and w0 w0 w w0 P0 : if the orbit of P0 does not intersect the pruning region Pw then P0 forces P0 . For example, a consequence of theorem 22 is the following corollary.

0 0 0 Corollary 26. Let w and w be two maximal decorations satisfying w >1 w and wb >1 wc. Then w w0 P0 >2 P0 .

0 0 w0 Proof. From conditions w >1 w and wb >1 wc, it holds that Pw ⊂ Pw0 and then Orb(P0 ) ∩ Pw = ∅. w w0 Thus, by theorem 22, P0 >2 P0 .  It is interesting to compare corollary 26 with one-dimensional dynamics where, by the Milnor- 0 ∞ Thurston kneading theory [37], the condition w >1 w implies that w010 forces the existence of w0010∞. This result is also closely related to [31, Theorem 4.8] by Holmes and Whitley where it was 2 0 proved that, for small values of b > 0 in the H´enonfamily Ha,b(x, y) = (a − x − by, x), if w and w 0 w w0 are maximal codes and wb01 >1 wc01 then P0 appears after P0 . Example 27. By corollary 26, it follows that

∞ 0 m n 0 ∞ ∞ 0 l k 0 ∞ 01110 1a10 1110 >2 01110 1b10 1110 m n 0 l k 0 for m > l > k, m > n > k and any words a and b such that 10 1a10 11 and 10 1b10 11 are maximal codes.

3.3. Concatenations of NBT decorations. Another group of decorations for which we know their pruning regions are these ones called concatenations of NBT codes. To define this kind of orbits we need the notion of NBT orbits introduced in [26] by T. Hall.

1 Definition 28. To every rational number q in Qb := Q ∩ (0, 2 ) we associate a symbol code in the m following way: If q = n , let Lq be the straight line segment joining the origin (0, 0) and the point 2 (n, m) in R . Then construct a finite word cq = s0s1...sn by the following rule:  1 if L intersects some line y = k, k ∈ , for x ∈ (i − 1, i + 1) (2) s = q Z i 0 otherwise

µ1 2 µ2 2 2 µm−1 2 µm It implies that cq is palindromic and has the form cq = 10 1 0 1 ··· 1 0 1 0 1. The (n+2)- periodic orbit Pq, having cq0 or cq1 as code, is called a NBT orbit. The following is the main result of [26] which claims that the forcing order restricted to NBT orbits coincides with the unimodal order.

Theorem 29 (Hall). Let q, q0 ∈ Qb. Then (i) Pq is quasi-one-dimensional, that is, Pq >1 R =⇒ Pq >2 R. 0 (ii) q ≤ q ⇐⇒ Pq >1 Pq0 ⇐⇒ Pq >2 Pq0 .

n A decoration w is a concatenation of NBT codes if there exists a finite sequence {qi}i=1 of distinct rational numbers in Qb such that

(3) w = cq1 0cq2 0 ··· 0cqn . DYNAMICS FORCED BY HOMOCLINIC ORBITS 23

We will give conditions for constructing a pruning diffeomorphism ψ relative to the homoclinic orbit w ∞ ∞ P0 , whose code is 01 · 0cq1 0cq2 0 ··· cqn 010 , with a well-defined pruning region. At first we have to organize the rational numbers in the real line qi1 > qi2 > ··· > qin . By theorem 29, their codes satisfy c 0 c 0 ··· c 0. Denote c = c = 10∞. qi1 61 qi2 61 61 qin q0 qn+1 w ∞ Definition 30 (Limiting points). Let Ci be the point of the orbit P0 whose code is 010cq1 0 ··· 0cqi−1 0· ∞ cqi 0 ··· 0cqn 010 . A point Ci, with i = 1, ··· , n, is a limiting point if there are no points Ck in the region ∞ ∞ cqi 0 ··· cqn 010 61 s+ 61 10 , Rii = {x = s− · s+ ∈ Σ2 : ∞ ∞ }. 0cqi−1 0cqi−2 ··· 0cq1 010 61 s− 61 010

The successor of a limiting point Ci is another limiting point Cj such that qj < qi and the region

∞ ∞ cqi 0 ··· cqn 010 61 s+ 61 10 , Rij = {x = s− · s+ ∈ Σ2 : ∞ ∞ } 0cqj−1 0cqj−2 ··· 0cq1 010 61 s− 61 010 does not contain any other point Ck.

Figure 20. The list in (a) is a P-list, but this one in (b) is not.

Denote by L the set of all the limiting points. It follows that C1 is always a limiting point and we ∞ ∞ will consider that Cn+1 = 010cq1 0 ··· 0cqn 0 · 10 is a limiting point as well. The limiting points in

figure 20(a) are C1, C4 and C5 and the filled region is R44. The successors of C1 and C4 are C4 and

C5, respectively. In figure 20(b) the limiting points are C1,C2, C3 and C5 and successors of C1,C2 and C3 are C3, C5 and C2, respectively. The filled region is R32.

n Definition 31 (P-list). We say that {qi}i=1 is a P-list if for every limiting point Ci, with 1 ≤ i ≤ n, and its successor Cj it holds that i < j ≤ n + 1 and the points Ck, for all i < k < j, are not limiting points. When the successor of Ci is Cn+1, we require that Ck, for all i < k ≤ n, do not be limiting points.

All lists with 2 or 3 elements are P-lists. The list in figure 20(a) is a P-list. For the list in figure

20(b), since the successor of C1 is C3, but C2 is also a limiting point, it follows that it is not a P-list. n When {qi}i=1 is a P-list, there exists a pruning region defined as follows.

n Definition 32 (Pruning domains). Let {qi}i=1 be a P-list. For every limiting point Ci and its i successor Cj define the domain Di which is bounded by a stable segment θs containing the homoclinic i ∞ point Ci and an unstable segment θu passing through the periodic point Ti = (cqi 0 ··· 0cqj−1 1) . The i i ∞ ∞ end-points of θs and θu are the heteroclinic points (cqi 0cqi+1 0 ··· cqj−1 1) · cqi 0 ··· 0cqn 010 and ∞ ∞ (cqi 0cqi+1 0 ··· cqj−1 1)cqi 0cqi+1 0 ··· cqj−1 0 · cqi 0 ··· 0cqn 010 . 24 VALENT´IN MENDOZA

n Definition 33 (Pruning region). The pruning region of the list {qi}i=1 or of the decoration w = cq1 0 ··· 0cqn is the set [ (4) Pw = Int(Di). Ci∈L Now one can prove the following result.

n Theorem 34. Let {qi}i=1 be a P-list and let w = cq1 0cq2 ··· 0cqn be its associated concatenation. Then C w = {x ∈ Σ : Orb(x,F ) ∩ P = ∅} up to a finite-to-one semi-conjugacy which is injective on the P0 2 w set of non boundary periodic points.

Proof. Let Cj be the successor of C1 and consider the domain D1 as in definition 33 which is 1 s 1 u bounded by a stable segment θs ⊂ W (C1) and an unstable segment θu ⊂ W (T1) where T1 = ∞ M1 (cq1 0cq2 0 ··· 0cqj−1 1) . Let M1 be the period of T1. It is useful to note that D1 and F (D1) have M+1 the same behaviour as the domains D1 and F (D1) for maximal decorations. See figure 21(a) for a geometrical explanation of these facts. 1 To see that D1 is a pruning domain note that the end-points of θs are the heteroclinic points ∞ ∞ A1 = (cq1 0cq2 0 ··· cqj−1 1) · cq1 0 ··· 0cqn 010 and

∞ ∞ A2 = (cq1 0cq2 0 ··· cqj−1 1)cq1 0cq2 0 ··· cqj−1 0 · cq1 0 ··· 0cqn 010 .

l l0 l l0 l l0 The iterates of A1 and A2 can be of the forms: ··· 0 11 · 0 , ··· 0 · 110 ··· , ··· 0 1 · 10 ··· , ··· cqk ·

0cqk+1 ··· and ··· cqk 0·cqk+1 ··· with 1 < k < j. The four forms to the left are clearly disjoint from D1. n 1 Since Ck is not a limiting point then the fifth form is disjoint from Int(D1). Thus F (θs )∩Int(D1) = ∅ for all n ≥ 1. As in the proof of theorem 22, it implies that D1 is a pruning domain.

(a) (b)

M1 Figure 21. (a) The first pruning domain D1 and its iterate f (D1). (b) The second pruning domain defined by the successor Cj.

Let ψ1 be the pruning Smale map associated constructed using D1. Thus ψ1 has a hyperbolic basic i set Kψ1 such that Kψ1 = {x ∈ Σ2 : F (x) ∈/ Int(D1)} up to a finite-to-one semiconjugacy. As in the case of maximal decorations (proposition 24), it follows that Kψ1 has an unstable periodic point with code B1,j1B1,j0B1,j1 where B1,j = cq1 0 ··· cqj−1 . Again as for maximal decorations it can be proved ∞ ∞ that ψ1 has a bigon I1 situated to the right of Cj, and formed by a stable segment of 0 · 10 and an ∞ unstable segment of (B1,j1B1,j0B1,j1) . See figure 21(b). DYNAMICS FORCED BY HOMOCLINIC ORBITS 25

Figure 22. Pruning region Pw associated to a concatenation of NBT orbits and the invariant manifolds of its pruning diffeomorphism ψ.

Now the proof continues pruning ψ1 by using the domain Dj which is defined by Cj and its successor i s j u ∞ 0 Cj , that is, Dj is bounded θs ⊂ W (Cj) and θu ⊂ W (Tj) where Tj = (cqj 0 ··· 0cqj0 1) . Since the invariant manifolds of the points of Kψ1 are just deformations of the invariant manifolds of the Smale horseshoe (proposition19) it is sufficiente to prove that Tj ∈ Kψ1 because in that case it maintains its invariant manifolds and then Dj is still a domain for ψ1.

Since {qi} is a P-list, as it was proved for T1, one can conclude that Orb(Tj) does not intersect

Int(D1) and Int(Dj). Hence Tj ∈ Kψ1 and Dj is a pruning domain. Thus we can construct the pruning diffeomorphism ψj associated to Dj. Proceeding in the same way with all the limiting points we will arrive to a pruning diffeomorphism ψ w i with a basic set Kψ and without bigons relative to P0 . Moreover Kψ = {x ∈ Σ2 : F (x) ∈/ Pw, ∀i ∈ Z} up a finite-to-one semiconjugacy, where P = ∪ Int(D ). By theorem 14, C w = K up a finite- w Ci∈L i P0 ψ to-one semiconjugacy which is injective on the set of non-boundary periodic points. See figure 22. 

If {qi} is not a P-list then some iterate of some Ti belongs to a pruning domain Dk. It implies that the unstable manifolds in Dk are deformed before of making the pruning isotopy in Dk. So we have no more control on these invariant segments and then the proof above can not be implemented. As in the case of maximal decorations, there exists a description of the boundary periodic points of

Kψ.

Proposition 35. Let ψ be the pruning map associated to a P-list {qi} and let Ci be a limiting point whose successor is Cj. Then the periodic orbit B with code Bi,j1Bi,j0Bi,j1, where Bi,j = cqj 0 ··· 0cqj−1 , ∞ ∞ is an u-boundary periodic orbit of ψ. The unstable manifolds of (Bi,j1Bi,j0Bi,j1) , (Bi,j0Bi,j1Bi,j1) ∞ and (Bi,j1Bi,j1Bi,j0) define a 3 sides principal region that contains a 3-pronged periodic singularity with code Bi,j1.

w Example 36. Consider the homoclinic orbit P0 defined by the P-list {2/5, 2/7, 1/3} with code ∞ ∞ ∞ ∞ 010c2/50c2/70c1/30 · 10 = 010101101010011001010010 · 10 . By theorem 34, its pruning region is formed by the union of the interior of the following two pruning domains: D1 bounded by a segment of ∞ ∞ the stable manifold of the limiting point C1 = 010·c2/50c2/70c1/3010 and a segment of the unstable ∞ ∞ manifold of T1 = (c2/50c2/71) = (1011010100110011) , and D2 bounded by a stable segment of the ∞ ∞ ∞ ∞ limiting point C3 = 010c2/50c2/70 · c1/3010 and an unstable segment of T2 = (c1/31) = (10011) . These domains are represented in figure 23.

A direct consequence of theorem 34 is a relation between decorations with the same combinatorics. 26 VALENT´IN MENDOZA

w Figure 23. Pruning region Pw associated to the homoclinic orbit P0 = ∞ ∞ ∞ ∞ 010c2/50c2/70c1/30 · 10 = 010101101010011001010010 · 10 , and the set of orbits forced by it.

n 0 n 0 0 Definition 37. Two P-lists {qi}i=1 and {qi}i=1 have the same combinatorics if qi < qj ⇐⇒ qi < qj. So we can prove the following.

n 0 n 0 Corollary 38. Let {qi}i=1 and {qi}i=1 be two P-lists with the same combinatorics. If qi < qi for all i = 1, ··· , n then

c 0 0···0cq0 cq1 0···0cqn q1 n P0 >2 P0 .

c 0 0···0c 0 0 cq 0···0cqn q qn Proof. Let {Di} and {Di} be the pruning domains associated to P 1 and P 1 , respec- 0 tively. Since {qi} and {qi} have the same combinatorics then there exists the same number of pruning 0 domains for each homoclinic orbit. By hypothesis q < q which implies that c 0 > c 0 0. By definition i i qi 1 qi

c 0 0···0cq0 0 0 q1 n of Di and Di it follows that Di ⊂ Di and so Orb(P0 ) ∩ P = ∅, where P is the pruning region

cq1 0···0cqn of P0 . So the conclusion follows from theorem 34.  3.4. Star homoclinic orbits. In [44] Yamaguchi and Tanikawa have dealt star homoclinic orbits

q ∞ ∞ ∞ µ1 2 µ2 2 2 µm−1 2 µm ∞ P0 which have as codes 0 · cq0 = 0 · 10 1 0 1 ··· 1 0 1 0 10 , with q ∈ Qb. These orbits have received this name because their train track types are star [17, 32]. Here we will see that they have well-defined pruning regions. To define it, construct the domain Dq bounded by s 2 ∞ ∞ u ∞ a segment θs ⊂ W (σ ( 0 · cq0 )) and a segment θu ⊂ W (1 ) which intersect at the heteroclinic points ∞1·0µ1−1120µ2 12 ··· 120µm−1 120µm 10∞ and ∞10·0µ1−1120µ2 12 ··· 120µm−1 120µm 10∞. See figure

24(a). Using the properties of cq, we can prove that Pw = Int(Dq) is a pruning domain associated q q to P0 and that its pruning diffeomorphism does not have bigons relative to P0 . Hence we have the following result.

Proposition 39. C q = {x ∈ Σ : Orb(x,F ) ∩ Int(D ) = ∅} up to a finite-to-one semi-conjugacy P0 2 q which is injective on the set of non boundary periodic points.

0 Noting that q ≥ q if and only if Dq ⊂ Dq0 , one have the following consequence of proposition above.

Proposition 40. [44, Theorem 5.2.1] For star homoclinic orbits, the forcing order agrees with the 0 0 q q order of the rational numbers, that is, q ≥ q ⇐⇒ P0 >2 P0 . As in proposition 24, we can prove that the basic set of the pruning diffeomorphism ψ associated ∞ to Dq has a boundary periodic orbit Rq of period n whose code is (ceq1) , where ceq is cq dropping the DYNAMICS FORCED BY HOMOCLINIC ORBITS 27 last two symbols. That orbit Rq defines a principal region of n sides which contains the fixed point 1∞.

2/7 ∞ ∞ ∞ ∞ Example 41. Consider q = 2/7. Then P0 has as code 0 · c2/70 = 0 · 100110010 . Its pruning 2 ∞ ∞ ∞ ∞ region D2/7 is defined by a stable segment of σ ( 0 · c2/70 ) = 010 · 0110010 and an unstable segment of the fixed point 1∞, and it is represented in figure 24(a) until its fourth iterate. In figure 2/7 24(b) we have represented the periodic orbits with periods less than 17 which are forced by P0 .

2/7 ∞ ∞ Figure 24. (a) Pruning region Dq of the homoclinic orbit P0 = 0 · 100110010 . 2/7 (b) The set of orbits of periods less than 17 which are forced by P0 .

Example 42. As an example of application to a concrete system, here we show a H´enonmap with 2 a well-defined pruning region. Consider the H´enon map H0,1.11(x, y) = (−x − 1.11y, x). Numerically the structure of the invariant manifolds of this map is similar to the structure of the pruning map 1/5 00 ∞ ∞ associated to the star homoclinic orbit P0 = P0 = 010000 · 10 . See figure 25(a). The pruning domain associated to that orbit is the domain D1/5 depicted in figure 25(b) which is bounded by s ∞ ∞ u ∞ θs ⊂ W ( 010 · 00010 ) and θu ⊂ W (1 ). Hence the orbits of H0,1.11, represented in figure 25(c), 00 correspond to the points forced by P ,Σ 00 = {x ∈ Σ : Orb(x,F )∩Int(D ) = ∅} up a finite-to-one 0 P0 2 1/5 semi-conjugacy.

00 (a) H´enonmap H0,1.11(x, y) (b) Pruning domain (c) Orbits forced by P0

00 ∞ ∞ Figure 25. Homoclinic orbit P0 = 010000.10 , its pruning region in the symbol square 00 and the set of orbits forced by P0 . 28 VALENT´IN MENDOZA

In the examples showed in previous subsections, there exists a well-defined pruning region obtained applying the differentiable pruning theorem a finite number of times. It is not true for all decorations, in fact, there exist decorations w for which their pruning regions are formed by an infinite number of pruning domains. But, even in that case, our method says that C w is the set of orbits that do not P0 intersect those pruning domains except for a certain type of boundary periodic orbits. See [35] for examples and details.

4. A method for eliminating bigons of a Smale map

In the following suppose that f is an orientation preserving Smale map on D2 having a saddle set K. From section2, to solve the problem of finding orbits forced by a homoclinic orbit P , we just have to find a Smale map without bigons relative to P . In this section we will introduce the pruning method that will allow us to eliminate the bigons of an arbitrary Smale map relative to a homoclinic orbit. This method generalizes the applications, given in section3, of the differentiable pruning theorem on homoclinic orbits of the Smale horseshoe. We will prove the following theorem.

Theorem 43. Let f be a Smale diffeomorphism having a wandering or non-wandering bigon I and let P ⊂ K be an orbit (periodic or homoclinic) disjoint from I. Then there exists a pruning domain containing I and disjoint from P .

Next subsections are devoted to the proof of theorem 43. We will prove it only for wandering bigons, since minor modifications can state the same conclusion for non-wandering bigons. Now we will give the details.

4.1. Finding the maximal domain. Let I be a wandering bigon with ∂I = αs ∪ αu and let P be a periodic or a homoclinic orbit to a fixed point p. This section describes how to find the maximal domain, relative to P , which contains I. The ideia consists in finding the maximal region D which is a region obtained enlarging the bigon I to a bigger non-wandering bigon. Such region, which is bounded by a stable segment θs and an unstable segment θu, contains I and a rectangle R, that is, a region for which there exists a homeomorphism h : [0, 1] × [0, 1] → R such that: s •R∩ W (K) = h(Fs × [0, 1]), where Fs is a closed subset of [0, 1] and {0, 1} ⊂ Fs, u •R∩ W (K) = h([0, 1] × Fu), where Fu is a closed subset of [0, 1] and {0, 1} ⊂ Fu. • D ∩ K = R ∩ K. s u Note that there are two possibilities for D: (a) D is bounded by θs ⊂ W (ps) and θu ⊂ W (pu) s where ps and pu are s-boundary and u-boundary periodic points, respectively and (b) θs ⊂ W (r) and u θu ⊂ W (r) where r is both s-boundary and u-boundary periodic point. See figure 26.

If an element pj of the orbit P lies in Int(D) then we consider the domain Dj which is bounded s s by a stable segment θs ⊂ W (pj) (in the case P is periodic) or θs ⊂ W (p) ( in the case that P is homoclinic) which contains pj. See figure 27. Thus the maximal domain D is the smallest of all these domain Dj. In all the cases or a piece of θu bounds a region E that is wandering and whose boundary is formed by n stable segments and n unstable segments with n ≥ 3, or θu contains a boundary period point. See figure 26. If the maximal domain is a pruning domain then theorem 43 is proved. Thus it remains to consider the case when D is not a pruning domain. DYNAMICS FORCED BY HOMOCLINIC ORBITS 29

Figure 26. The maximal domain D containing I is bounded by the invariant seg- ments θu and θs.

Figure 27. The maximal domain D containing I is bounded by the invariant seg- ments θu and θs.

4.2. Defining a pruning domain from D. If the maximal region D is not a pruning domain one 0 must prove that it is always possible to decrease θs until a sub-segment θs ⊂ θs which defines a pruning 0 0 0 domain D , that contains I, whose boundary consists of θs and an unstable segment θu. We need the following lemma.

n Lemma 44. If f (θs) ∩ Int(D) = ∅, for all n ≥ 1 then D satisfies pruning conditions for f.

n ˚ −n ˚ Proof. By hypothesis f (θs) ∩ θu = ∅ for all n ≥ 1 then f (θu) ∩ θs = ∅ for all n ≥ 1. So if there −n −n exists n ≥ 1 such that f (θu) ∩ Int(D) 6= ∅ then f (θu) ⊂ Int(D). This is not possible since a piece −n of θu is in the boundary of E or θu contains a boundary periodic point r and, then f (E) ⊂ Int(D) or f −n(r) ⊂ Int(D), which is a contradiction with the definition of D. So D satisfies the pruning conditions of definition 16. 

Ni By lemma 44 we just need to study the case when there exist positive integers Ni such that f (θs)∩ Int(D) 6= ∅. 30 VALENT´IN MENDOZA

N Lemma 45. Let D be the maximal domain and an integer N > 1. If f (θs) ∩ Int(D) 6= ∅ then N ˚ N f (θs) ∩ θu 6= ∅ or αs ⊂ f (θs).

Proof. Let γs be the stable segment such that αs ⊂ γs and the end-points of γs are included in θu. N ˚ s N u If f (θs) ⊂ (W (K) ∩ D) \ γs, then f (θs) can be continued through the leaves of W (K) ∩ D. Finding the N-backward iterate of these continuation, we can see that the rectangle defining D can be extended to a bigger one. That is a contradiction with the definition of D. The same argument N N N works if f (θs) ⊂ γs \ αs. The remaining case is when f (θs) ∩ I= 6 ∅ that implies αs ⊂ f (θs). N ˚ s N ˚ If f (θs) is not included in W (K) ∩ D then f (θs) ∩ θu 6= ∅. 

Ni Figure 28. If the maximal domain is not a pruning domain, an iterate f (θs) of θs intersects Int(D) as in the Figure.

Actually there is only a finite number of positive integers satisfying conditions of lemma above.

Ni Lemma 46. There are a finite number of positive integers N1,N2, ..., Nl such that f (θs)∩Int(D) 6= ∅, for all i = 1, ..., l.

Ni Proof. Suppose that there exists a sequence {Ni}i∈N of positive integers such that f (θs)∩Int(D) 6= ∅. Ni See figure 28. By lemma 45 and since diam(f (θs)) goes to 0 when i goes to ∞, it follows that only a

Ni Ni ˚ finite number of the Ni satisfy f (θs) ⊃ αs. So we just have to consider the case when f (θs)∩θu 6= ∅. s 0 0 Since θs ⊂ W (ps), there exist a subsequence {Nk} ⊂ {Ni} and a point ps of the orbit of ps such that N 0 0 0 0 lim f i (θs) goes to ps when i goes to ∞, and then one have that ps ∈ θu. Hence ps is a periodic point u 0 0 n in W (pu) which is only possible if ps = pu ∈ θs ∩ θu and ps = ps. But in this case f (θs) ∩ Int(D) 6= ∅ for a finite number of positive integers. That is a contradiction.  For the iterates of lemma 46, we have:

Lemma 47. For every Ni there exists a subdomain Di ⊂ D such that ∂Di = θs,i ∪θu,i where θs,i ⊂ θs,

Ni f (θs,i) ∩ Int(Di) = ∅ and θu,i is an unstable leaf included in D.

Ni Ni Proof. Since f (θs)∩Int(D) 6= ∅ one see that the projection of f (θs) along the unstable leaves of D intersects θs. Thus one can decrease θs until a small segment θs,i such that (a) θs,i has its end-points

Ni in the same unstable leaf θu,i and (b) f (θs,i) projects along the unstable leaves and intersects θs,i at only one point. See figure 29.  With the preceding lemmas we can prove now theorem 43.

Proof of theorem 43. If the maximal domain D of section 4.1 satisfies the pruning conditions then N theorem 43 is proved. If D does not satisfy pruning conditions then by, lemma 46, f (θs)∩Int(D) = ∅ DYNAMICS FORCED BY HOMOCLINIC ORBITS 31

Figure 29. Decreasing the domain D to another Di.

for all positive integer N ≥ 1 except for a finite number of positive integers Ni with i = 1, ··· , l. From lemma 47 for each Ni there exists a subdomain Di ⊂ D with boundary θs,i ∪ θu,i such that θs,i ⊂ θs and θu,i is an unstable segment joining the end-points of θs,i. l Since θs,i ⊂ θs,j or θs,j ⊂ θs,i, for all i, j ∈ {1, .., l}, the domains {Di}i=1 can be organized by 0 0 inclusion. Let i be the positive integer such that θs := θs,i0 is the smallest segment in {θs,i}, so the 0 0 domain D := Di0 is the smallest of all the domains D1, ··· ,Dl. Let θu be the unstable segment joining 0 0 0 0 Ni 0 0 the end-points of θs. So ∂D = θs ∪ θu and f (θs) ∩ Int(D ) = ∅, for all i = 1, ··· , l. Hence

n 0 0 (5) f (θs) ∩ Int(D ) = ∅, for all n ≥ 1. Therefore for proving that D0 is a pruning domain it is sufficient to prove that

−n 0 0 (6) f (θu) ∩ Int(D ) = ∅, for all n ≥ 1.

0 Ni0 0 0 0 By construction of D , it follows that f (θu) ∩ θu 6= ∅ and so by [39, Theorem 1.2], θu is included

0 Ni0 0 in the unstable manifold of a periodic point qu. If A is the end-point of θs such that f (A) ∈ θu

0 Figure 30. The unstable piece θu contains a periodic point qu.

Ni0 0 0 then, since f preserves orientation, one have that f (θu) contains A. See figure 30. Decreasing θu it 0 0 follows that there exists a periodic point in θu which, by uniqueness, has to be qu. Hence qu ∈ θu. 0 n ˚0 0 Now we will prove that D satisfies condition (6). Since f (θs) ∩ θu = ∅, for all n ≥ 1, then ˚0 −n 0 −n 0 0 −n 0 0 θs ∩ f (θu) = ∅, for all n ≥ 1. It implies that if f (θu) ∩ Int(D ) 6= ∅ then f (θu) ⊂ Int(D ). −N 0 0 0 Suppose that there are positive integers N such that f (θu) ⊂ Int(D ). Since θu has a periodic −1 0 0 point qu of period Ni and f is contracting on θu, there is a finite number of positive integers Mij

−Mi 0 0 N 0 0 0 0 j i with Mij < Ni , j = 1, .., k such that f (θu) ⊂ Int(D ). Recall that f (θs) intersects θu at 0 0 Ni0 −Mij +Ni −Mij +Ni 0 0 the point f (A). So f (A) ∈ f (θs) ∩ Int(D ). It is a contradiction with (5) since 0 0 −Mij + Ni > 0. Therefore the domain D , associated to the bigon I, satisfies pruning conditions which proves theorem 43.  32 VALENT´IN MENDOZA

4.3. Pruning Method. Now it is clear how the pruning method works.

Algorithm 48. Given an orbit (periodic or homoclinic) P of a Smale map f then: (1) If f has a bigon relative to P , applying theorem 43 to f and P , we can find a pruning domain

D1. Then theorem 18 allows us to construct a pruning diffeomorphism f1.

(2) If f1 has no bigons relative to P then, by theorem 14, every periodic orbit of f1 is forced by P .

(3) If f1 has bigons then return to (1).

(4) Repeating (1), (2) and (3) we will obtain a sequence f1, f2 ··· . If this sequence is finite

{f1, ··· , fn} then, by theorem 14, the orbits of the basic set of fn are forced by P up a finite-to-one semiconjugacy which is injective on the set of periodic orbits.

5. Acknowledgements

This research was partially supported by FAPESP grant 2010/20159-6. I would like to thank the institute IME-USP for the hospitality during the time in which parts of this work was done. I am grateful to Andr´ede Carvalho for his stimulating conversations about pruning, to Toby Hall that has written the useful programs Prune and Train to explore the pruning regions and the train track of periodic horseshoe points, and to Philip Boyland who talk us about isotopy stability during his stay at IME-USP. I would also thank the referee of the first draft of this manuscript who made many helpful comments for improving some unclear aspects contained in the proofs.

References [1] S. K. Aranson and V.Z. Grines. Homeomorphisms with minimal entropy on two-dimensional manifolds. Russian Math. Surveys 36:2 (1981), 163–198. [2] S. K. Aranson and V.Z. Grines. The topological classification of cascades on closed two-dimensional manifolds. Russian Math. Surveys 45:1 (1990), 1–35. [3] D. Asimov and J. Franks, Unremovable closed orbits, in Geometric Dynamics (Rio de Janeiro, 1981), Lectures and Notes in Math. 1007. Springer, Berl´ın,(1983), 22–39. [4] F. Beguin, C. Bonatti and B. Yu. Building Anosov flows in three-manifolds. Geom. Topol. 21(3) (2017), 1837- 1930. [5] C. Bonatti, R. Langevin and E. Jeandenans, Diff´eomorphismesde Smale des surfaces. Soci´et´eMath´ematiquede France (1998). [6] P. Boyland, Topological methods in surface dynamics. Topology and its Applications 54 (1994), 223–298. [7] P. Boyland, Isotopy stability of dynamics on surfaces. Contemp. Math. 246 (1999), 17–46. [8] P. Boyland and T. Hall, Isotopy stable dynamics relative to compact invariant sets. Proc. London Math. Soc. 79(3) (1999), 673–693. [9] J. Cantwell and L. Conlon, Hyperbolic geometric and homotopic homeomorphisms of surfaces. Geometriae Dedicata 177(1) (2015), 27–42. [10] J. Cantwell and L. Conlon, Endperiodic automorphisms of surfaces and foliations. arXiv:1006.4525v6 (2017). [11] P. Collins, Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits. Dyn. Syst., 19(1) 1–39 (2004). [12] P. Collins, Entropy-minimizing models of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, Dyn. Syst. 20(4) 369–400 (2005). [13] P. Cvitanovic´, Periodic orbits as the skeleton of classical and quantum chaos. Physica D 51 138–151 (1991). [14] A. de Carvalho, Pruning fronts and the formation of horseshoes. Ergod.Th. and Dynam. Sys. 19(4), 851–894 (1999). [15] A. de Carvalho, Extensions, quotients and generalized pseudo-Anosov maps, In Graphs and patterns in mathe- matics and theoretical physics, volume 73 of Proc. Sympo. Pure Math., pages 315–338, Amer. Math. Soc., Providence, RI, (2005). DYNAMICS FORCED BY HOMOCLINIC ORBITS 33

[16] A. de Carvalho and T. Hall, The forcing relation for horseshoe braid types. Experiment. Math. 11(2), 271–288 (2002). [17] A. de Carvalho and T. Hall, Braid forcing and star-shaped train tracks. Topology 43, 247–287 (2004). [18] A. de Carvalho and V. Mendoza, Differentiable pruning and the hyperbolic pruning front conjecture. Preprint (2014). [19] A. Fathi, F. Laudenbach and V. Poenaru´ , Travaux de Thurston sur les surfaces. Asterisque 66-67 (1979). [20] S. Fenley, End periodic homeomorphisms and 3-manifolds. Mathematische Zeitschrift 224, 1–24 (1997). [21] R. W. Ghrist, P. Holmes and M. C. Sullivan, Knots and links in three-dimensional flows. Lectures Notes in Mathematics. 1654 (1997). [22] V. Z. Grines, On the topological equivalence of one-dimensional basic sets of diffeomorphisms of two-dimensional manifolds. Uspekhi Mat. Nauk 29(6) (1974) 163–164. [23] V. Z. Grines, A representation of one-dimensional attractors of A-diffeomorphisms by hyperbolic homeomorphisms. Mathematical Notes 62(1) (1997) 64–73. [24] V. Z. Grines, T. V. Medvedev, O. Pochinka, Dynamical systems on 2 and 3 manifolds. Part of the Developments in Mathematics book series (DEVM, volume 46). Springer International Publishing, 2016. [25] T. Hall, Unremovable periodic orbits of homeomorphisms. Math. Proc. Camb. Phil. Soc. 110, 523–531 (1991). [26] T. Hall, The creation of horseshoes. Nonlinearity. 7, 861–924 (1994). [27] M. Handel, Global shadowing of pseudoanosov homeomorphisms. Ergod.Th. and Dynam. Sys. 5, 373–377 (1985). [28] M. Handel. A fixed-point theorem for planar homeomorphisms. Topology. 38(2) (1999) 235–264. [29] M. Handel and R. T. Miller, End periodic surface homeomorphisms. Unpublished manuscript. [30] H. M. Hulme, Finite and infinite braids: a dynamical systems approach. Ph.D. thesis (Univ. of Liverpool, 2000). [31] P. Holmes and D. Whitley, Bifurcations of one- and two-dimensional maps. Phil. Trans. R. Soc. Lond. A. 311(1515), 43–102 (1984). [32] E. Kin and P. Collins, Templates forced by the Smale horseshoe trellises (developments and applications of dynamical systems theory), RIMS Kokyuroku 1369 204–211 (2004). [33] J. Lewowicz and R. Ures, On basic pieces of Axiom A diffeomorphisms isotopic to pseudoanosov maps. Asterisque, 287, 125–134 (2003). [34] J. Los, On the forcing relation for surface homeomorphisms. Publications math´ematiquesde l’I.H.E.S.´ 85, 5–61 (1997). [35] V. Mendoza, Ordering periodic horseshoe orbits: Pruning models and forcing. Preprint. (2014). [36] V. Mendoza, Domains: software available at https://www.dropbox.com/s/w701quf3viooycm/Domains.exe [37] J. Milnor and W. Thurston, On iterated maps of the Interval. Dynamical Systems (Lectures Notes in Mathematics Vol. 1342). Berlin:Springer. 465–563 (1988). [38] S. Newhouse and J. Palis, Hyperbolic nonwandering sets on two-manifolds, Dynamical Systems, ed. M. Peixoto, Acad. Press, 293–301 (1973). [39] R. V. Plykin, Sources and sinks of A-difeomorphisms of surfaces. Jour. Math. USSR-Sb. 23(2) 233–253 (1974). [40] S. Smale, Differentiable dynamical systems. Bull. Amer. Math. Soc., 73, 747–817 (1967). [41] K. Tanikawa and Y. Yamaguchi, Forcing relations for homoclinic orbits of the reversible horseshoe map. Progr. Theoret. Phys. 126(5) 811–839 (2011). [42] R. F. Williams, The DA maps of Smale and structural stability. Global Analysis, Proc. of Symp. Pure Math., 14, 329–334 (1970). [43] Y. Yamaguchi and K. Tanikawa, Order of appearance of homoclinic points for the H´enonmap, Progr. Theoret. Phys. 116(6) 1029–1049 (2006). [44] Y. Yamaguchi and K. Tanikawa, A new interpretation of the symbolic codes for the H´enonmap, Progr. Theoret. Phys. 122(3) 569–609 (2009).

Instituto de Matematica´ e Computac¸ao,˜ Universidade Federal de Itajuba,´ Av. BPS 1303, Bairro Pinheir- inho, CEP 37500-903, Itajuba,´ MG, Brazil. E-mail address: [email protected]